Abstract

Recently, a beautiful paper of Andrews and Sellers has established linear congruences for the Fishburn numbers modulo an infinite set of primes. Since then, a number of authors have proven refined results, for example, extending all of these congruences to arbitrary powers of the primes involved. Here, we take a different perspective and explain the general theory of such congruences in the context of an important class of quantum modular forms. As one example, we obtain an infinite series of combinatorial sequences connected to the ‘half-derivatives’ of the Andrews-Gordon functions and with Kashaev’s invariant on (2m+1,2) torus knots, and we prove conditions under which the sequences satisfy linear congruences modulo at least 50% of primes.

Introduction and statement of results

In his seminal 2010 Clay lecture, Zagier defined a new class of functions with certain automorphic properties called ‘quantum modular forms’ [39]. Roughly speaking, these are complex-valued functions defined on the rational numbers which have modular transformations modulo ‘nice’ functions. Although the definition is (intentionally) vague, Zagier gave a handful of motivating examples to serve as prototypes. For example, he defined quantum modular forms related to Maass cusp forms attached to Hecke characters of real quadratic fields, as studied by Andrews, Dyson, and Hickerson [2] and Cohen [14], and he gave examples related to sums over quadratic polynomials and non-holomorphic Eichler integrals. More precisely, Zagier made the following definition.

Definition.

A quantum modular form is a function f:P1(Q)→C for which f(x)−f|kγ(x) is ‘suitably nice’.

Here |k is the usual Petersson slash operator, and ‘suitably nice’ means that the obstruction to modularity satisfies an appropriate analyticity condition, e.g., Ck,C∞, etc. One of the most striking examples of a quantum modular form is described in Zagier’s paper on Vassiliev invariants [40], in which he studies Kontsevich’s function F(q) given by

F(q):=∑n=0∞(q)n,

(1)

where (a)n:=(a,q)n:=∏j=0n−11−aqj.

This function does not converge on any open subset of , but converges as a finite sum for q any root of unity. Zagier’s study of F depends on the ‘sum of tails’ identity

∑n≥0η(q)−q124qn=η(τ)Dq−12∑n≥1nχ12(n)qn2−124,

(2)

where η(q):=q124(q)∞, D(q):=−12+∑n≥1qn1−qn, and χ12(n):=12n. Recalling the identity (q)∞=∑n≥1χ12(n)qn2−124, we find that the last term on the right-hand side of (2) is essentially a ‘half-derivative’ of η. Zagier further observed that η(q) and η(q)D(q) vanish to infinite order as q approaches a root of unity. Thus, we have what Zagier terms a ‘strange identity’ of the shape

F(q)“=′′−12∑n≥1nχ12(n)qn2−124.

(3)

Although the left- and right-hand sides of (3) do not ever converge simultaneously, as we will see below, this identity can be interpreted as saying that there is an equality between asymptotic expansions when q=e−t as t→0+ (a similar statement holds as q approaches other roots of unity as well).

Since [39], there has been an explosion of research aimed at constructing examples of quantum modular forms related to Eichler integrals, extending the initial applications to knot invariants and quantum invariants of three-manifolds in [28] and [40]. For instance, such quantum modular forms are closely tied to surprising identities relating generating functions of ranks, cranks, and unimodal sequences [20]; are related to probabilities of integer partition statistics [35]; and arise in the study of negative index Jacobi forms and Kac-Wakimoto characters [7]. For further examples, see also [5],[10],[11],[16],[24],[26],[31],[36].

The general theory of this class of quantum modular forms was further elucidated by Choi, Lim, and Rhoades in [18], and from a different perspective by Bringmann and the third author in [9], where the space of ‘Eichler quantum modular forms’ was defined. In particular, for each half-integral weight cusp form, there is an associated Eichler integral with quantum modular properties. At the end of [9], a program was laid out to study the general properties of these quantum modular forms. In particular, one of the fundamental problems in the theory was identified to be the determination of the arithmetic properties of such forms. This problem was inspired by recent work of Andrews and Sellers [4] in which they studied the congruence properties of the Fishburn numbers defined by

∑n≥0ξ(n)qn:=∑n≥01−q;1−qn.

These numbers are important in combinatorics and knot theory, and in particular, ξ(n) enumerates the number of linearly independent Vassiliev invariants of degree n[40]. For related results on the conbinatorics and asymptotic analysis of the Fishburn numbers and related sequences, see also [3],[6],[8],[12],[13],[19],[27],[30],[37]. Utilizing beautiful and clever manipulations, Andrews and Sellers proved an infinite family of congruences for the Fishburn numbers, and these results were extended in several directions by Garvan [21], Straub [38], and Ahlgren and Kim [1]. For example, the work of Andrews and Sellers and Straub implies that for all A,n∈N, we have

ξ5An−1≡ξ5An−2≡0mod5A,

ξ7An−1≡0mod7A,

ξ11An−1≡ξ11An−2≡ξ11An−3≡0mod11A.

In the context of quantum modular forms, the Fishburn numbers are the coefficients of the power series expansion at the root of unity q=1 of the Eichler quantum modular form associated to η. In light of this connection, Garvan and Straub posed the following natural question.

Question(Garvan, Straub).

What is the general theory of such congruences for more general quantum modular forms?

In this paper, we answer this question for a class of quantum modular forms. In particular, we consider quantum modular forms which arise as Eichler integrals of unary theta series. That is, we consider the quantum modular forms associated to theta series of the form

∑n∈Zχ(n)nνqn2−a2b,

where ν∈{0,1}, χ is a periodic sequence of mean value zero, and a,b∈Z. For the sake of definiteness, we will focus our attention on the case when ν=0, i.e., when the unary theta series has weight 1/2. We will shortly see how to define a ‘Fishburn-type’ sequence associated to any such unary theta series via a series expansion as q→1. We remark in passing that the analogous study of congruences for general modular forms of half-integral weight does not make sense as we do not know any algebraicity results about the associated power series coefficients.

Before stating our main result, we define a condition on the periodic sequences χ under consideration. This condition arises very naturally from the perspective of p-adic measures below and is also satisfied by our canonical family of examples arising from knot invariants (see the ‘Fishburn numbers and Hikami’s functions’ section). Specifically, our condition on χ is as follows, where

ψu(x):=1ifx≡u(modM),0otherwise,

and ψu,v(x):=ψu(x)−ψv(x).

Definition.

Let χ:Z→{0,1,−1} be a periodic function with period M and mean value zero. We say that χ is a good function if it is a sum of functions of the form ψu,v such that there exist natural numbers C:=C(u,v) with (C,M)=1 and uC≡v (mod M).

Remark.

The condition for χ to be good is trivially satisfied whenever χ is supported on integers coprime to M. In particular, any quadratic Dirichlet character is good.

To describe our main result, we first consider for any a,b∈Z and any periodic sequence χ the following partial theta function:

Pa,b,χ(q):=∑n≥0nχ(n)qn2−a2b.

Note that this is essentially a half-derivative of the series ∑nχ(n)qn2−a2b. From the general theory outlined in [9], χ has mean value zero in particular whenever Pa,b,χ is a half-derivative of a modular form which is cuspidal at q=1. From now on, we will always assume that Pa,b,χ∈Z[[q]]. As we will see in the proof of Theorem 4, the mean value zero property of χ implies that there is an asymptotic expansion

Pa,b,χe−t∼∑n≥0αa,b,χ(n)tn

for some αa,b,χ(n)∈C as t→0+. Thus, we may define coefficients Ha,b,χ(n) by the relation

Pa,b,χe−t∼∑n≥0Ha,b,χ(n)1−e−tn.

It is easily seen that the coefficients Ha,b,χ(n) are defined by finite recursions in terms of αa,b,χ(n). Moreover, the explicit formulas for the coefficients in Theorem 4 show that Ha,b,χ(n)∈Q for all n. We will see below that in important examples arising from knot theory, they may also be defined combinatorially in a manner similar to the definition of the Fishburn numbers above.

Our main result is then the following, where we define βp=β to be the coefficient of p1 in the base p expansion of −a2b.

Theorem1.

Let χ be a good function and suppose that a,b∈Z are chosen so that Pa,b,χ∈Z[[q]]. Then, for any prime p not dividing b, the following are true.

(1)

If B is a positive integer such that

a2−bp=−1,a2−2bp=−1,…,a2−Bbp=−1,

then for all A,n∈N, we have

Ha,b,χpAn−B≡0modpA.

(2)

If β≠(p−1) and B is a positive integer such that

a2−bp≠1,a2−2bp≠1,…,a2−Bbp≠1,

then for all A,n∈N, we have

Ha,b,χpAn−B≡0modpA.

Three remarks.1) We will see in the ‘Fishburn numbers and Hikami’s functions’ section that Theorem 1 implies all known linear congruences for ξ(n). 2) We will give a stronger version of Theorem 1 in Theorem 7 below. 3) As asked by Andrews and Sellers and Straub in the case of ξ(n), it is an interesting question to determine a converse of Theorem 1 classifying linear congruences for the sequences Ha,b,χ(n). This seems difficult to prove using our techniques. From the perspective of p-adic measures outlined in the ‘Preliminaries’ section, Theorem 3 aligns with the intuitive notion that the ‘integral’ of a function which is p-adically small over a compact set is small. However, the difficulty in proving a converse result lies in the fact that the integral of a large function can still be small.

In particular, we have the following corollary, as will be shown in the ‘Fishburn numbers and Hikami’s functions’ section, which implies that infinitely many of the sequences arising in our family of examples from knot theory satisfy linear congruences modulo at least 50% of primes.

Corollary2.

Let χ be a good function and suppose that a,b∈Z are chosen so that Pa,b,χ∈Z[[q]], and that a2−b≢3 (mod 4) and such that a2−b is not a square. Then, there is a linear congruence for Ha,b,χ(n) modulo at least 50% of primes.

Remark.

We will see in the ‘Fishburn numbers and Hikami’s functions’ section that the condition that a2−b is not a square is necessary, where we will give an example in which our theorem does not guarantee any congruences.

We now highlight an important situation when the coefficients Ha,b,χ(n) may be defined without using asymptotic expansions of partial theta functions. To describe this situation, we require the Habiro ring, which was brilliantly studied by Habiro in [22] and further connected to the F1 story and to the theory of Tate motives in [34]. This ring is defined as the completion

ℋ:=lim←n≥0Z[q]/((q)n),

which may, as usual, be realized as the set of formal expansions of the form

∑n≥0an(q)(q)n,

where an∈Z[q]. Associated to any element of the Habiro ring and to any root of unity ζ, there is a power series expansion in (ζ−q) [22]. That is, there is a map

ϕζ:ℋ→Z[ζ][[ζ−q]].

For example, the map ϕ1 may be realized explicitly (and be computed efficiently) for any F=∑n≥0an(q)(q)n∈ℋ as the expansion ∑n≥0cn1−qn by recursively solving for cn in the expansion

∑n≥0an(1−q)(1−q;1−q)n=∑n≥0cnqn.

The resulting expressions defining cn always terminate, since (1−q;1−q)n=O(qn). In particular, recalling the definition of ξ(n), the Fishburn numbers are the coefficients of ϕ1(F), where F is Kontsevich’s function.

A central result of Habiro (see Theorem 5.2 of [22]) states that the map ϕζ is injective for any ζ, so that in fact, an element of the Habiro ring is uniquely determined by its power series expansion at any given root of unity. As Habiro points out, this is similar to the fact that a holomorphic function is determined by its power series expansion at a single point, so that ℋ may be thought of as a ‘ring of analytic functions on the roots of unity’.

Returning to the question of congruences, we now suppose that Pa,b,χ satisfies a ‘strange identity’, i.e., that there is an Fa,b,χ∈ℋ such that

Pa,b,χe−t∼Fa,b,χe−t.

(4)

Defining ca,b,χ(n) as the coefficients of ϕ1(Fa,b,χ), it directly follows from (4) and the definition of Ha,b,χ(n) that

Ha,b,χ(n)=ca,b,χ(n).

Hence, we have given congruences for the coefficients of power series expansions of any element of the Habiro ring which satisfies a strange identity connecting it to a partial theta function Pa,b,χ satisfying the conditions of Theorem 1. In the ‘Fishburn numbers and Hikami’s functions’ section, we study the half-derivatives of Andrews-Gordon functions, which satisfy strange identities as shown by Hikami in [25]. Furthermore, the resulting elements of the Habiro ring arise naturally in the context of Kashaev’s invariant for the (2m+1,2) torus knots and include Kontsevich’s function as a special case.

More generally, it is very interesting to ask which partial theta functions satisfy strange identities, and what the general structure of such identities should be. For work related to this question, the interested reader is referred to the important work of Coogan, Lovejoy, and Ono in [15],[32], which studies connections between asymptotic expansions of partial theta series and q-hypergeometric series.

The paper is organized as follows. In the ‘Preliminaries’ section, we recall some standard facts on p-adic measures and L-functions, and we prove a fundamental result concerning congruences of polynomials in L-values, as well as a useful expression of the coefficients Ha,b,χ(n) in terms of L-values. Together, these results allow us to reduce the proof of Theorem 1 to an elementary statement on congruences for binomial coefficients, which we prove along with Corollary 2 in the ‘Proof of Theorem 1 and Corollary 2’ section. Finally, in the ‘Fishburn numbers and Hikami’s functions’ section, we give an illuminating example of importance in knot theory, which also shows that Theorem 1 yields all known congruences for the Fishburn numbers, as well as new congruences for a class of sequences naturally generalizing the Fishburn numbers.

Preliminaries

Congruences for polynomials in L-values

In this subsection, we prove a useful theorem regarding the p-adic properties of certain polynomials in L-values. For any periodic sequence χ(n) of mean value zero, we define for Re(s)≫0 the L-function

Lχ(s):=∑n≥1χ(n)ns.

This L-function has an analytic continuation to all of , which can easily be seen by writing L as a linear combination of specializations of the Hurwitz zeta function and using the well-known continuation for the Hurwitz zeta function, which only has a simple pole of residue one at s=1. Thus, in particular, the values of L(s) for s∈−N0 are well defined.

In order to state our theorem, we first define a linear operator Lχ:Q[x]→C by its action on generators as

Lχxn:=Lχ(−n),

and we recall that f∈Q[x] is called a numerical polynomial on a set X⊂Z if f(x)∈Z for all x∈X. We also denote by supp(χ) the support of χ. Our main result concerning congruences of L-values is as follows.

Theorem3.

Let χ be a good function with period M, p a prime with (p,M)=1, and A∈N. If f is a numerical polynomial on supp(χ) such that for all n∈supp(χ), we have

f(n)≡0modpA,

then we also have

Lχ(f)≡0modpA.

Remark.

This theorem is a specialization of fairly standard facts on p-adic Dirichlet L-functions, or, if one prefers, p-adic Mazur measures (see, e.g., [17], [23], Chapter 3]). However, for the reader’s convenience, we present a full and elementary proof below.

Proof.

We begin by recalling the definition of generalized Bernoulli numbers Bk,χ, which are given by the generating function

∑a=0M−1χ(a)teateMt−1=:∑k≥0Bk,χtkk!.

As mentioned above, Lχ has an analytic continuation to all of . At non-positive integers, this is realized by the well-known relation

Lχ(−n)=−Bn+1,χn+1.

Furthermore, it is possible (see [29], Chapter XIII, Theorem 1.2]) to present these numbers as p-adic limits of related power sums:

Bk,χ=limn→∞1Mpn∑a=0Mpn−1χ(a)ak.

We thus have that

Lχxk=Lχ(−k)=−Bk+1,χk+1=−limn→∞1Mpn∑a=0Mpn−1χ(a)ak+1k+1.

Now let c>1 be a positive integer such that (c,Mp)=1, and define

Lχc(f):=Lχ(f)−Lχcfc,

where the functions fc and χc are defined as fc(x):=f(cx) and χc(x):=χ(cx). We now write

f(x)=∑m=0Ndmxm

with dm∈Q and use the above to compute

Lχc(f)=−∑m=0Ndmlimn→∞1Mpn∑a=0Mpn−1χ(a)am+1m+1−χ(ca)(ca)m+1m+1.

Let ac be the element of {0,1,…Mpn−1} such that ac≡ca (mod Mpn). Since (c,Mp)=1, multiplication by c permutes residues modulo Mpn, and we can rearrange the sum as

Lχc(f)=−∑m=0Ndmlimn→∞1Mpn∑a=0Mpn−1χ(ca)acm+1−(ca)m+1m+1.

We now let ac=ca+Mpnta with ta∈Z, make use of the congruence

acm+1−(ca)m+1=(ca+Mpnta)m+1−(ca)m+1≡(m+1)(ca)mMpntamodMpn2,

and reorder the summations to conclude that

Lχc(f)=−limn→∞∑a=0Mpn−1taχ(ca)f(ca).

In particular, we derive that for every c>1 such that (c,Mp)=1,

Lχc(f)≡0modpA

whenever f(n)≡0(mod pA) for every n∈supp(χ).

However, this does not suffice since we need the congruence for Lχ(f) itself, not merely for the modified version Lχc(f). Since χ is a good function, it suffices to prove the congruence for the function ψu,v(n) where u and v satisfy the conditions of the definition of a good function. We assume that u and v do satisfy these conditions for the remainder of the proof. Now note that for an integer c such that (c,Mp)=1, we have

ψuc(x)=ψu(cx)=ψuc−1(x),

where cc−1≡1 (mod M). We may now choose two integers c1>1 and c2>1 satisfying

(c1,Mp)=(c2,Mp)=1uc1≡v(modM)c2≡1(modM)c1≡c2(modpC)

for any large C, which we choose later.

We already know that

Lψvc1(f)=Lψv(f)−Lψu(fc1)≡0(modpA)

and

Lψuc2(f)=Lψu(f)−Lψufc2≡0(modpA),

and it suffices to prove that C can be chosen big enough to guarantee the congruence

Lψufc1−Lψufc2=Lψufc1−fc2≡0(modpA).

However, the assumptions on u and v imply that

Lψufc1−fc2=∑m=0Ndmc1m−c2mBm+1,ψum+1

for some rational quantities Bm+1,ψu/(m+1), which makes our claim obvious.

A useful formula for the sequence Ha,b,χ(n)

In this subsection, we relate the values of the sequences Ha,b,χ(n) to polynomials in L-values. Namely, we show the following result.

Theorem4.

Assume the conditions of Theorem 1. Then, we have

Ha,b,χ(n)=(−1)nLχxx2−a2bn.

Proof.

Plugging in q=e−t into the definition of Pa,b,χ(q), we find

Pa,b,χe−t=∑n≥1nχ(n)e−n2−a2bt.

Using a shifted version of the Euler-Maclaurin summation formula or a simple Mellin transform argument (see, for example, the proposition on page 98 of [28]), we find as t→0+ the asymptotic expansion

Proof of Theorem 1 and Corollary 2

We begin with a result giving a family of congruences for the sequences Ha,b,χ(n) which we will shortly see implies Theorem 1. Firstly, however, we derive an elementary lemma on congruences of binomial coefficients. Specifically, we now recall Kummer’s theorem, which allows us to easily study such congruences.

Let p be a prime, and suppose n∈Z, k∈N. Then, the p-adic order of nk equals the number of carries when adding k to n−k in base p.

From this, one can easily obtain the following lemma (see also Lemma 3.4 of [38]).

Lemma6.

Let p be a prime, s∈{0,1,…,p−1}, and α∈N. Then, the following are true.

(1)

If B∈{1,2…,p−s−1}, then for all A,n∈N,

s+pαpAn−B≡0(modpA).

(2)

If B∈{1,2…,p−1} and α≢−1 (mod p), then for all A,n∈N,

s+pαpAn−B≡0(modpA−1).

Proof.

Write n:=s+pα and k:=pAn−B, and denote the base p coefficients of n (resp. k) by n0,n1,… (resp. k0,k1,…). By assumption, n0=s and k0=p−B. Moreover, we have k1=k2=…=kA−1=p−1. We now split into cases.

Proof of (1): As B∈{1,2…,p−s−1}, we have k0>n0. Denoting the base p coefficients of n−k by m0,m1,…, we find that m0=p−k0+n0, so that there is a carry when m0 is added to k0. Since k1=k2=…=kA−1=p−1, there are at least A carries occurring when k is added to n−k, so by Theorem 5, we find the desired congruence.

Proof of (2): By part (1) of the Lemma, we may suppose that B∈{p−s,p−s+1,…,p−1}, i.e., that k0≤n0. Note that m0=n0−k0, and by the assumption on α, we find n1≠p−1, so that m1>0. Hence, when adding k to n−k, a carry occurs when adding m1 to k1. Since k2=k3=…=kA−1=p−1, there is also a carry when mi is added to ki for i=2,…A−1, so that there are at least A−1 carries when k is added to n−k. By Theorem 5, the result follows.

In order to state our generalized version of Theorem 1, we first set

Sa,b,χ,p:=S:=s∈N0:s<p,∃x∈supp(χ),x≢0(modp),x2−a2b≡s(modp),

and we define an analogous set

Sa,b,χ,p∗:=S∗:=s∈N0:s<p,∃x∈supp(χ),x2−a2b≡s(modp).

Then, our main result is as follows.

Theorem7.

Let χ be a good function and p a prime, and suppose a,b∈Z are chosen so that Pa,b,χ∈Z[[q]]. Then, the following are true.

(1)

If B∈{1,2,…,p−1−maxs∈S∗s}, then for all n,A∈N, we have

Ha,b,χpAn−B≡0(modpA).

(2)

If β≠(p−1) and (b,p)=1, then for any B∈{1,2,…,p−1− maxs∈Ss} and for all n,A∈N, we have

Ha,b,χpAn−B≡0(modpA).

Proof.

We begin by splitting into cases. Proof of (1): By Theorem 3 and Theorem 4, it suffices to show that for B∈{1,2,…,p−1−maxs∈S∗s}, we have

xx2−a2bpAn−B≡0(modpA)

for all A,n∈N, x∈supp(χ). For fixed x, let y:=x2−a2b, and let s∈N be the reduction of y modulo p with 0≤s<p. We claim that we can choose α∈N so that

ypAn−B≡s+pαpAn−B(modpA).

For this, it is enough to choose α satisfying

(y)pAn−B≡(s+pα)pAn−B(modpA+C),

where C:= ordp((pAn−B)!). Clearly, it suffices to choose α with

y≡s+pα(modpA+C).

Now we can set y−s=:zp, where z∈Z. The last equation is then equivalent to

α≡z(modpA+C−1),

which is clearly possible to solve for α. Thus, for such an α, we have

ypAn−B≡s+pαpAn−B(modpA).

Now (1) of Lemma 6 directly implies that

s+pαpAn−B≡0(modpA)

for all B∈{1,2,…,p−1− maxs∈S∗s}, n∈N, x∈supp(χ).

Proof of (2): To complete the proof, it suffices to show that if x≡0 (mod p), β≠(p−1), and (b,p)=1, then for all 0≤B≤p−1n∈N, we have

xx2−a2bpAn−B≡0(modp).

In this case, we find that y≡−a2b(modp2). Hence, we may choose α as above and assume without loss of generality that y≡s+pα (mod p2). As β≠(p−1), we have that α≢−1 (mod p). By (2) of Lemma 6, we find that

s+pαpAn−B≡0(modpA−1).

Hence,

xx2−a2bpAn−B≡xs+pαpAn−B≡0(modpA),

as desired.

We are now in a position to prove Theorem 1. First, consider the sets

Ta,b,χ,p:=T:=s∈N0:s<p,∃x∈Z,x≢0(modp),x2−a2b≡s(modp),

Ta,b,χ,p∗:=T∗:=s∈N0:s<p,∃x∈Z,x2−a2b≡s(modp).

Clearly, since S⊂T, and S∗⊂T∗ if B satisfies the conditions of Theorem 7 with S replaced by T or with S∗ replaced by T∗, then the same congruence for Ha,b,χ holds. Theorem 1 follows from this observation, together with the following elementary result, whose proof follows from a straightforward calculation.

Lemma8.

Let a,b∈Z, let p be a prime and suppose (p,b)=1. Then, the following are equivalent conditions for B∈N.

(1)

We have that B satisfies

a2−bp=−1,a2−2bp=−1,…,a2−Bbp=−1.

(2)

We have that

B∈1,2,…,p−1−maxt∈T∗t.

Moreover, the following are also equivalent conditions for B∈N.

(1)

We have that B satisfies

a2−bp≠1,a2−2bp≠1,…,a2−Bbp≠1.

(2)

We have that

B∈1,2,…,p−1−maxt∈Tt.

Remark.

An elementary argument shifting integers x by multiples of p shows that if (p,M)=1, then we have S=T and S∗=T∗. Hence, for all but finitely many primes p, the congruences given in Theorem 7 are exactly the same as the congruences implied by Theorem 1.

We now prove Corollary 2.

Proof (Proof of Corollary 2).

It suffices to check that if a2−b is not a square and a2−b≢3 (mod 4), then exactly 50% of primes p satisfy a2−bp=−1, since Theorem 1 then implies that Ha,b,χ(pn−1)≡0 (mod p) for all n∈N. To see that this is the case, note that given the conditions on a2−b, the function a2−b· is a non-principal Dirichlet character and hence takes on values 1 and −1 equally often (as the sum over a complete set of representatives of residue classes modulo the modulus of the character is zero). Moreover, the values where this character are non-zero are exactly those values which are coprime to the modulus of the character. Hence, by the Chebatorev density theorem applied to primes in arithmetic progressions, exactly 50% of primes satisfy the desired condition.

Fishburn numbers and Hikami’s functions

In this section, we work out a particularly important family of examples of Theorem 1. Specifically, we consider a collection of quantum modular forms whose beautiful properties were laid out by Hikami in [25]. For further important results ‘inverting’ these functions and relating them to indefinite theta series and mock theta functions, see also [11],[26]. The quantum modular forms defined by Hikami are then given for m∈N and α∈{1,2,…,m−1} by

We note that the function F1(0)(q) reduces simply to Kontsevich’s function F(q). Moreover, it is clear from the definition that Fm(α)∈ℋ. The connection to partial theta functions is shown in (15) of [25], which states that Fm(α) shares an asymptotic expansion with a half-derivative of an Andrews-Gordon function. Specifically, Hikami shows that there is a strange identity connecting Fm(α) and

Moreover, by the discussion in the ‘Introduction and statement of results’ section, in this case, the coefficients H2m−2α−1,8(2m+1),χ8m+4(α) may be defined combinatorially as the coefficients of the expansion

∑n≥0ξm(α)(n)qn:=Fm(α)(1−q),

so that

H2m−2α−1,8(2m+1),χ8m+4(α)(n)=ξm(α)(n).

As an example, consider

F2(0)(q)=∑n≥0(q)n∑k=0nqk(k+1)nkq,

so that the first few coefficients ξ2(0)(n) are given by 1,2,6,23,109,…. Numerical calculations suggest that the following congruences hold for all n,A∈N:

ξ2(0)3An−1≡0(mod3A),

ξ2(0)11An−a≡0(mod11A),wherea∈{1,2,3},

ξ2(0)13An−a≡0(mod13A),wherea∈{1,2,3,4}.

These congruences follow immediately from Theorem 1 and Lemma 8, along with a short computation, once the following result is checked.

Lemma9.

For any m∈N, a∈{1,2,…,m−2}, χ8m+4(α)(n) is a good function.

Proof.

Let x:=2m+1 and y:=α+1. Then, the period of χ8m+4(α) is M=4x, and by the definition of the sequence, it is clearly supported on odd integers. Furthermore, one easily checks that

χ8m+4(α)=ψx−2y,x+2y−ψ3x−2y,3x+2y.

Now it is easy to verify that

(4x,x−2y)=(4x,x+2y)=(x,y)

and

(4x,3x−2y)=(4x,3x+2y)=(x,y).

Hence, ψx−2y,x+2y and ψ3x−2y,3x+2y both satisfy the conditions of a good function, so that χ8m+4(α) is good as well.

Thus, we may apply Theorem 1 to the coefficients ξm(α), which directly implies congruences for ξm(α)(n). Using Corollary 2 and the fact that

2m−2α−12−8(2m+1)≡0,1(mod4),

we immediately deduce the following result.

Corollary10.

Choose α,m∈N with α<m such that (2m−2α−1)2−8(2m+1) is not a square. Then

ξm(α)pAn−1≡0(modpA)

for all n,A∈N for at least 50% of primes p.

Remark.

The condition that (2m−2α−1)2−8(2m+1) is not a square is necessary in Corollary 10. For example, if α=1, m=7, then we have a2=121, b=120. Hence, Theorem 1 yields a congruence modulo p only when a2−bp=1p≠1, which does not hold for any p.

We now take a closer look at the congruences for ξ(n), which inspired this paper. Using the well-known fact that for p≥5, p2−124∈Z, we find that if p2−124≡s(modp) with 0≤s<p, then

βp=p2−124−sp=p2−124p≠(p−1).

By Theorem 1, we find that

ξpAn−1≡0(modpA)

for all A,n∈N whenever

−23p≠1.

Using quadratic reciprocity, it is easy to check that this occurs exactly when

Finally, we show that our set of congruences for ξ(n) is the same as that which was given in [38]. Indeed, one easily finds by an elementary argument that the set S in Theorem 7 may be replaced by the set T (since (p,M)=1). Comparing with Theorem 1.2 of [38], it suffices to show that the set of reductions of values of x2−124 modulo p as x ranges over Z∖pℤ is equal to the set of reductions of pentagonal numbers 12x(3x−1) modulo p with x≢0 (mod p) for any prime p≥5. However, this follows immediately by noting that x2−124 becomes 12x(3x−1) upon substituting x with 6x−1, which simply permutes the residue classes of x modulo p. Thus, thanks to the extensive calculations of Straub [38], we have conjecturally given all linear congruences for ξ(n). That is, following Andrews and Garvan and Straub, we make the following conjecture, which we leave as an important challenge for future work.

Conjecture.

Let p be a prime. Then, there exists a B∈N such that there is a congruence

ξpAn−B≡0(modpA)

for all n precisely when

p=23orp≡5,7,10,11,14,15,17,19,20,21,22(mod23).

We conclude by noting that there are other congruences for linear combinations of Fishburn numbers. Specifically, in unpublished work, Garthwaite and Rhoades observed that for all n∈N, we have

ξ(5n+2)−2ξ(5n+1)≡0(mod5),

ξ(11n+7)−3ξ(11n+4)+2ξ(11n+3)≡0(mod11).

In Theorem 1.3 of [21], Garvan established an infinite set of congruences which includes this example. It is likely that the methods of this paper extend to prove these congruences of Garvan and that similar congruences hold for other quantum modular forms. We leave the details to the interested reader.

Declarations

Acknowledgements

The research of the first author is supported by the Simons Foundation Collaboration Grant. This research was conducted while the first author was a guest at MPIM, and he is grateful to the Institute for making this research possible. The first author thanks Don Zagier for an elucidating communication. The third author is grateful for the support of the DFG through a University of Cologne postdoc grant.

Copyright

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